Sample average approximation for the continuous type principal-agent problem

A Bayesian approach to continuous type principal-agent problems

“…Here, we propose a different approach relative to Singham (2019). Specifically, we provide a general Bayesian approach to the problem using Markov Chain Monte Carlo (MCMC) to provide statistical inferences.…”

“…In turn, the agent chooses the quantity and price option depending on his / her private demand. Singham (2019) provides a method for bounding the optimal profit and finding solution estimates for the continuous version of the principal-agent problem. This method can also be used when the density f (θ) is not simple enough to yield analytical solutions, or even when f (θ) is not available but can be estimated using a sample of random numbers from the corresponding distribution.…”

“…For an example in principal-agent problems, see where a sample-average approximation is used (in the context of an example). More general models are considered in Singham (2019). The principal's objective is to offer the agent a contract {q(θ), τ (θ)} which specifies a quantity q(θ) to be delivered at total price τ (θ) for each given demand.…”

“…In this problem, (IR) is the individual rationality constraint, (IC) is the incentive compatibility constraint, and (MON) is the monotonicity constraint. Singham (2019) observed that if θ (m) , m = 1, ..., M is a random sample from the distribution whose pdf is f (θ), and θ (1) ≤ . .…”

“…Here, q m ≡ q(θ (m) ) and ∆ m ≡ ∆(q(θ (m) ) are decision variables defined as discrete points in the contract function. Singham (2019) proposed a bootstrapping approximation based on lower and upper bounds for the objective function and showed that the algorithm works well in practice. In this paper, we cast the problem in terms of Bayesian inference and show that use of the bootstrap as in Singham ( 2019) can be avoided.…”